📘 Fundamental Theorem of Calculus (FTC I & II)

🔹 FTC Part I (Derivative of Integral)

If $F(x) = \int_a^x f(t)\,dt$, then:

$$F'(x) = f(x)$$

Key Idea:

• Differentiation undoes integration

With chain rule:

If $F(x) = \int_a^{g(x)} f(t)\,dt$, then:

$$F'(x) = f(g(x)) \cdot g'(x)$$

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🔹 FTC Part II (Evaluation of Definite Integral)

If $F'(x) = f(x)$, then:

$$\int_a^b f(x)\,dx = F(b) - F(a)$$

Steps:

1. Find an antiderivative $F(x)$
2. Plug in bounds: $F(b) - F(a)$

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🔹 Quick Example

$$\int_1^3 x^2\,dx$$

Antiderivative: $F(x) = \frac{x^3}{3}$

$$= \frac{3^3}{3} - \frac{1^3}{3} = \frac{27}{3} - \frac{1}{3} = \frac{26}{3}$$

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🔹 Key Takeaways

• FTC connects derivatives ↔ integrals
• Part I: derivative of area function
• Part II: compute definite integrals
• Always check if chain rule is needed